Existence of Solution of the Nonlinear Differential Equation in the Modelling of Eardrum by Using Homotopy Perturbation Method

Praveen, J. Peter; Rao, B. N.; Harnath, Y.; Rao, Bopeng; Narayana, C.; Mahaboob, B.

doi:10.37418/amsj.9.7.51

Cited by 5 publications

(4 citation statements)

References 5 publications

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“…As we mentioned above, there are a numerous number of authors who have investigated the (un)damped Duffing equation [49,50], (un)damped Helmholtz equation [29,[51][52][53][54], and the undamped H-D equation [55,56]. On the other side, there is a few numbers of published papers that were devoted for deriving some approximate solutions to the damped H-D equation [57][58][59][60].…”

Section: ̈mentioning

confidence: 99%

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Novel solutions to the (un)damped Helmholtz-Duffing oscillator and its application to plasma physics: Moving boundary method

2021

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Novel analytical and numerical solutions to the (un)damped Helmholtz-Duffing (H-D) equation for arbitrary initial conditions are derived. Both the analytical (for undamped case) and approximate analytical (for damped case) solutions, are obtained in the form of Weierstrass elliptic function. Also, the soliton solution to the undamped H-D equation is obtained in detail. The semi-analytical solution to the damped H-D equation is compared to the fourth-order Runge-Kutta (RK4) numerical solution. The obtained solution shows an excellent agreement with the numerical simulations but sometimes (according to the values of initial conditions) not on all time interval. Thus, the moving boundary method is utilized to improve the semi-analytical solution. It is found that the improved solution gives good results with high accuracy in the whole time domain. As a realistic application, the obtained solutions are applied to the study of nonlinear oscillations in an electronegative non-Maxwellian dusty plasma. Finally, we conclude that our novel solutions could help us to understand the dynamics of various nonlinear oscillations in engineering and in different branches of sciences such as oscillations in different plasma models.

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Section: ̈mentioning

confidence: 99%

“…If the frictional force is ignored, some of these equations can be converted into the undamped Duffing or Helmholtz equation or the undamped H-D equation. For instance, the mKdV equation ¶ F + F ¶ F + ¶ F = ( ) ( ) [29]. However, the Gardner equation…”

Section: Introductionmentioning

confidence: 99%

Novel solutions to the (un)damped Helmholtz-Duffing oscillator and its application to plasma physics: Moving boundary method

2021

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“…Recently, many scientists and researchers have applied HPM to solve numerous problems which arise in science, technology, and health, such as nonlinear problems arising in heat transfer [19], the nonlinear Burgers' equation [20], the fractional KdV-Burgers' equation [21], the Helmholtz-Fangzhu oscillator [22], linear parabolic equations [23], the Balitsky-Kovchegov equation [24], a one-dimensional convection diffusion problem [25], a fractional wave equation [26], a shock wave equation [27], linear fuzzy delay differential equations [28], a mathematical model of dengue fever [29], a mathematical model of the eardrum [30], a mathematical model of mumps [31], systems of second-order nonlinear ordinary differential equations [32], iterative methods for nonlinear equations [33], and a model of depletion of forest resources [34]. Agarwal et al [35] applied the HPM and Bernoulli equation to measure the COVID-19 outbreak.…”

Section: Introductionmentioning

confidence: 99%

Study of the Six-Compartment Nonlinear COVID-19 Model with the Homotopy Perturbation Method

Rafiullah,

Asif,

Jabeen

et al. 2024

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The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by S(t), E(t), A(t), I(t), R(t), and B(t), respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the (n−1)th-degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods.

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“…Perturbation and asymptotic approximations are generally applicable for weakly nonlinear ODEs (ordinary differential equations) and PDEs (partial differential equations) having small/large physical parameters. Khatami et al [1] have successfully applied the DTM (differential transform method) and obtained solutions for nonlinear Duffing oscillators. The modified DTM provides inconsistency in the periodic solutions of the nonlinear Duffing oscillators having asymmetric oscillations [2][3][4].…”

Section: Introductionmentioning

confidence: 99%

On the applicability of HAM to seek periodic solution for truly nonlinear oscillator

2020

IJETER

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Homotopy perturbation method (HPM) is claimed to be a simple analytic approximation method suitable for solving nonlinear differential equations. It assures solution series convergence by transferring the nonlinear problem into a number of linear sub-problems. A differential equation which is really nonlinear oscillator is considered for obtaining periodic solution using the HPM. To examine the adequacy of HPM, the phase diagram made from the approximate solution is studied with respect to the actual phase diagram.

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Existence of Solution of the Nonlinear Differential Equation in the Modelling of Eardrum by Using Homotopy Perturbation Method

Cited by 5 publications

References 5 publications

Novel solutions to the (un)damped Helmholtz-Duffing oscillator and its application to plasma physics: Moving boundary method

Novel solutions to the (un)damped Helmholtz-Duffing oscillator and its application to plasma physics: Moving boundary method

Study of the Six-Compartment Nonlinear COVID-19 Model with the Homotopy Perturbation Method

On the applicability of HAM to seek periodic solution for truly nonlinear oscillator

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